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Properties

Label 16.0.20396718160...5841.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{8}\cdot 29^{12}$
Root discriminant $33.06$
Ramified primes $7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15451, 14611, 71142, -45695, 114255, -80505, 82259, -44509, 29440, -11821, 5845, -1676, 666, -126, 41, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 41*x^14 - 126*x^13 + 666*x^12 - 1676*x^11 + 5845*x^10 - 11821*x^9 + 29440*x^8 - 44509*x^7 + 82259*x^6 - 80505*x^5 + 114255*x^4 - 45695*x^3 + 71142*x^2 + 14611*x + 15451)
 
gp: K = bnfinit(x^16 - 4*x^15 + 41*x^14 - 126*x^13 + 666*x^12 - 1676*x^11 + 5845*x^10 - 11821*x^9 + 29440*x^8 - 44509*x^7 + 82259*x^6 - 80505*x^5 + 114255*x^4 - 45695*x^3 + 71142*x^2 + 14611*x + 15451, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 41 x^{14} - 126 x^{13} + 666 x^{12} - 1676 x^{11} + 5845 x^{10} - 11821 x^{9} + 29440 x^{8} - 44509 x^{7} + 82259 x^{6} - 80505 x^{5} + 114255 x^{4} - 45695 x^{3} + 71142 x^{2} + 14611 x + 15451 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2039671816037671133025841=7^{8}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.06$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{19320} a^{12} - \frac{1}{6440} a^{11} - \frac{2809}{19320} a^{10} + \frac{2153}{19320} a^{9} - \frac{353}{3220} a^{8} - \frac{4631}{19320} a^{7} - \frac{4339}{9660} a^{6} + \frac{1609}{3864} a^{5} + \frac{109}{230} a^{4} - \frac{267}{1288} a^{3} - \frac{727}{6440} a^{2} - \frac{199}{420} a + \frac{7109}{19320}$, $\frac{1}{19320} a^{13} - \frac{443}{9660} a^{11} - \frac{239}{9660} a^{10} + \frac{803}{6440} a^{9} + \frac{2539}{19320} a^{8} - \frac{5183}{19320} a^{7} + \frac{7127}{19320} a^{6} - \frac{1783}{6440} a^{5} - \frac{1839}{6440} a^{4} + \frac{42}{115} a^{3} + \frac{1111}{3864} a^{2} - \frac{1033}{19320} a + \frac{1313}{6440}$, $\frac{1}{19320} a^{14} + \frac{13}{345} a^{11} - \frac{259}{2760} a^{10} + \frac{153}{920} a^{9} + \frac{37}{19320} a^{8} - \frac{1321}{6440} a^{7} + \frac{197}{552} a^{6} + \frac{419}{2760} a^{5} + \frac{29}{115} a^{4} + \frac{887}{2760} a^{3} - \frac{1439}{3864} a^{2} - \frac{353}{3864} a + \frac{2059}{9660}$, $\frac{1}{127289711280788981400} a^{15} + \frac{52176316012013}{9092122234342070100} a^{14} + \frac{581788782170521}{42429903760262993800} a^{13} + \frac{93746002769863}{5303737970032874225} a^{12} + \frac{6339728799095085973}{127289711280788981400} a^{11} - \frac{133529741879778037}{2157452733572694600} a^{10} + \frac{2299019500254187527}{21214951880131496900} a^{9} + \frac{77671158256790396}{2273030558585517525} a^{8} - \frac{7616700353294993551}{31822427820197245350} a^{7} - \frac{2962782790765623781}{15911213910098622675} a^{6} - \frac{45154961406328892729}{127289711280788981400} a^{5} - \frac{31692827929135597997}{63644855640394490700} a^{4} - \frac{50222066157921661529}{127289711280788981400} a^{3} + \frac{26262151117432063943}{63644855640394490700} a^{2} - \frac{8738622874310743329}{42429903760262993800} a - \frac{18524244058930802231}{127289711280788981400}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 419388.064919 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-203}) \), \(\Q(\sqrt{-7}, \sqrt{29})\), 4.2.170723.1 x2, 4.0.1195061.1 x2, 4.0.1421.1 x2, 4.2.5887.1 x2, 4.4.1195061.1, 4.0.24389.1, 8.0.1428170793721.2, 8.0.1698181681.1, 8.0.1428170793721.1, 8.4.1428170793721.1 x2, 8.0.29146342729.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$